91影视

For each pair of functions f and g and interval $[a,b]$in Exercises 41鈥�52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f$and $g$from $x=a\mathrm{to}x=b$

$f\left(x\right)=x^2-x-1$and $g\left(x\right)=5-x^2$,$\left[-2,3\right]$

Find the sum or quantity without completely expanding or calculating any sums.

Given$\underset{k=0}{\overset{25}{鈭�}}k=325$and$\underset{k=3}{\overset{28}{鈭�}}{\left(k-3\right)}^2=14,910$, find the value of$\underset{k=3}{\overset{25}{鈭�}}\left(k^2-5k+9\right)$.

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$鈭�3x^2\sin \left(x^3+1\right)dx$

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$鈭�\frac{6x}{3x^2+1}dx$

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value.

$\underset{n鈫�鈭�}{\lim }\underset{k=1}{\overset{n}{鈭�}}\frac{k^2+k+1}{n}$

For each pair of functions $f$and $g$and interval $[a,b]$in Exercises 41鈥�52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f$and $g$from x = a to x = b.

width="330" height="23" role="math">f(x)=x-1,g(x)=x2-2x-1,[a,b]=[-1,3]

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_2^4\left(x^2+1\right)dx$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises 19鈥�64. Use

a graph to check your answer. (Hint: The integrands that involve

absolute values will have to be considered piecewise.)

${鈭�}_1^{e-3}\frac{3\mathrm{蟺}}{2x+6}dx$

Integral Formulas: Fill in the blanks to complete each of the

following integration formulas.

(The last six formulas involve hyperbolic functions and their inverses.)
$鈭�\frac{1}{\sqrt{x^2+1}}dx=\_\_\_\_$

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.

$\frac{d}{dx}{鈭�}_x^{x+2}鈥�\sin 鈦�\left(t^2\right)dt$